The Nyquist sampling theorem is a basis for modern communication and signal processing. The theorem regulates that in a process of analog/digital signal conversion, when a sampling frequency is no less than double of a highest frequency of a signal, the sampled digital signal completely retains information in the original signal. In general, in practical applications, it is ensured that the sampling frequency is 5-10 times of the highest frequency of the signal.
A broadband signal has a high frequency, and if the signal is directly sampled using the Nyquist sampling theorem, a very high sampling frequency is required. However, the existing analog-to-digital converter has a highest frequency of 1 GHz, which is difficult to satisfy the requirements for high frequencies. In addition, after the signal is processed by the analog-to-digital converter, the signal is generally stored for data communication and processing later. In this case, high-speed sampling puts high requirements for a storage speed of a memory and a processing speed of a digital signal processor. Therefore, if the broadband signal is sampled using the Nyquist sampling theorem, enormous pressures are brought to collection, storage, transmission and processing of the broadband signal.
Further, the Nyquist sampling theorem is developed to only use the minimum prior information of the signal to be sampled, i.e., a bandwidth of the signal, but does not use some structural characteristics of the signal per se, such as redundancy or the like.
In general, although the processed signal per se is not sparse, the signal may be represented in a sparse form under some transformation bases. Compressive sensing, as an emerging theorem, is to sample the signal with a frequency much lower than the Nyquist frequency by utilizing the sparse characteristics of the signal. The compressive sensing theorem regulates that as long as a high-dimensional signal can be represented in a sparse form under some transform domain, the high-dimensional signal can be projected into a low-dimensional space by using an observation matrix independent of the transformation base matrix, and then the original high-dimensional signal can be reconstructed from a few projections by solving optimization problems.